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UNIVERSITI MALAYSIA PAHANG

BORANG PENGESAHAN STATUS TESIS

JUDUL:

SESI PENGAJIAN:________________

Saya ________________________________________________________________

(HURUF BESAR)

mengaku membenarkan tesis (Sarjana Muda/Sarjana /Doktor Falsafah)* ini disimpan di

Perpustakaan dengan syarat-syarat kegunaan seperti berikut:

1. Tesis adalah hakmilik Universiti Malaysia Pahang (UMP).

2. Perpustakaan dibenarkan membuat salinan untuk tujuan pengajian sahaja.

3. Perpustakaan dibenarkan membuat salinan tesis ini sebagai bahan pertukaran antara institusi

pengajian tinggi.

4. **Sila tandakan ( )

(Mengandungi maklumat yang berdarjah keselamatan

SULIT atau kepentingan Malaysia seperti yang termaktub

di dalam AKTA RAHSIA RASMI 1972)

TERHAD (Mengandungi maklumat TERHAD yang telah ditentukan

oleh organisasi/badan di mana penyelidikan dijalankan)

TIDAK TERHAD

Disahkan oleh:

___________________________ ___________________________

(TANDATANGAN PENULIS) (TANDATANGAN PENYELIA)

Alamat Tetap:

D/A HIM MOTOR, KG PINTU MUHAMAD ZUHAIRI BIN SULAIMAN

GERBANG, JLN TAWANG, ( Nama Penyelia )

16020 BACHOK , KELANTAN

Tarikh: 20 NOVEMBER 2009 Tarikh: : 20 NOVEMBER 2009

CATATAN: * Potong yang tidak berkenaan.

** Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak

berkuasa/organisasi berkenaan dengan menyatakan sekali tempoh tesis ini perlu

dikelaskan sebagai atau TERHAD.

Tesis dimaksudkan sebagai tesis bagi Ijazah doktor Falsafah dan Sarjana secara

Penyelidikan, atau disertasi bagi pengajian secara kerja kursus dan

penyelidikan, atau Laporan Projek Sarjana Muda (PSM).

2009/2010

SITI AISYAH-AWANIS BINTI MOHD YUSOFF (870201-29-5546)

FINITE DIFFERENCE OF THERMAL LATTICE BOLTZMANN

SCHEME FOR THE SIMULATION OF NATURAL

CONVECTION HEAT TRANSFER

FINITE DIFFERENCE OF THERMAL LATTICE BOLTZMANN

SCHEME FOR THE SIMULATION OF NATURAL

CONVECTION HEAT TRANSFER

SITI AISYAH-AWANIS BINTI MOHD YUSOFF

Report submitted in partial fulfilment of the requirements

for the award of the degree of

Bachelor of Mechanical Engineering

Faculty of Mechanical Engineering

UNIVERSITI MALAYSIA PAHANG

NOVEMBER 2009

ii

SUPERVISOR’S DECLARATION

I hereby declare that I have checked this project and in my opinion, this project is

adequate in terms of scope and quality for the award of the degree of Bachelor of

Mechanical Engineering

Signature

Name of Supervisor: MUHAMAD ZUHAIRI BIN SULAIMAN

Position: LECTURER

Date: 20 NOVEMBER 2009

iii

STUDENT’S DECLARATION

I hereby declare that the work in this project is my own except for quotations and

summaries which have been duly acknowledged. The project has not been accepted for

any degree and is not concurrently submitted for award of other degree.

Signature

Name: SITI AISYAH-AWANIS BINTI MOHD YUSOFF

ID Number: MA06039

Date: 20 NOVEMBER 2009

iv

ACKNOWLEDGEMENT

In the name of Allah s.w.t, the most Gracious, the Ever Merciful Praise is to

Allah, Lord of the Universe and Peace and Prayers be upon His final prophet and

Messenger Muhammad s.a.w.

I am heartily thankful to my supervisor, Mr. Muhamad Zuhairi bin Sulaiman,

whose encouragement, guidance and support from the initial to the final level enabled

me to develop an understanding of this project. I also would like to express very special

thanks to my former supervisor, Mr. Mohd Rosdzimin bin Abdul Rahman for the

suggestions and co-operation throughout the study. I also sincerely thanks for the time

spent proofreading and correcting my many mistakes.

I reserve my sincere thanks for my family members. I am deeply indebted to my

father, Mohd Yusoff bin Mat Leh, my mother, Rabiah binti Abdullah, my sister, Siti

Mariam-Nabilah and my youngest sister, Siti Fatimah Najibah for their never ending

love, dedication, support and faith in me. Without them, this project would not been

done.

Special thanks should be given to my committee members. Lastly, I offer my

regards and blessings to all of those who supported me in any respect during the

completion of the project.

v

ABSTRACT

In this thesis, a method of lattice Boltzmann is introduced. Lattice Boltzmann

method (LBM) is a class of computational fluid dynamics (CFD) methods for fluid

simulation. Objective of this thesis is to develop finite difference lattice Boltzmann

scheme for the natural convection heat transfer. Unlike conventional CFD methods, the

lattice Boltzmann method is based on microscopic models and macroscopic kinetic

equation. The lattice Boltzmann equation (LBE) method has been found to be

particularly useful in application involving interfacial dynamics and complex

boundaries. First, the general concept of the lattice Boltzmann method is introduced to

understand concept of Navier-Strokes equation. The isothermal and thermal lattices

Boltzmann equation has been directly derived from the Boltzmann equation by

discretization in both time and phase space. Following from this concept, a few simple

isothermal flow simulations which are Poiseulle flow and Couette flow were done to

show the effectiveness of this method. Beside, numerical result of the simulations of

Porous Couette flow and natural convection in a square cavity are presented in order to

validate these new thermal models. Lastly, the discretization procedure of Lattice

Boltzmann Equation (LBE) is demonstrated with finite difference technique. The

temporal discretization is obtained by using second order Rungge-Kutta (modified)

Euler method from derivation of governing equation. The discussion and conclusion

will be presented in chapter five.

vi

ABSTRAK

Di dalam tesis ini, kaedah kekisi Boltzmann diperkenalkan. Kaedah kekisi

Boltzmann (LBM) ialah dikelaskan daripada kaedah pengiraan dinamik bendalir

berkomputer (CFD) untuk simulasi bendalir. Objektif tesis ini ialah untuk

membangunkan kaedah pembezaan terhingga kekisi Boltzmann untuk pemanasan

semulajadi pemindahan haba. Tidak seperti kaedah CFD, kaedah kekisi Boltzmann ialah

berdasarkan model mikroskopik dan persamaan kinetik makroskopik. Kaedah

persamaan kekisi Boltzmann ditemui untuk kegunaan terutamanya di dalam aplikasi

yang melibatkan hubungkait dinamik dan garisan sempadan yang rumit. Permulaannya,

konsep umum kaedah kekisi Boltzman diperkenalkan untuk memahami konsep

persamaan Navier-Strokes iaitu dengan diskritasikan persamaan Boltzmann terhadap

masa dan ruang fasa, persamaan untuk isoterma dan terma kekisi Boltzmann dapat

diterbitkan. Berdasarkan konsep ini, contoh yang mudah daripada pengaliran isoterma

ialah pengaliran Poiseulle dan pengaliran Couette telah dijalankan untuk menunjukkan

keberkesanan kaedah ini. Selain itu, keputusan berangka daripada simulasi pengaliran

Porous Couette dan pemanasan semulajadi di dalam ruang segiempat diperkenalkan

untuk mengesahkan model terma yang baru. Akhir sekali, diskritasikan prosedur

persamaan kekisi Boltzmaan diperkenalkan dengan menggunakan teknik pembezaan

terhingga. Hasil daripada pemberolehan persaman governing, masa diskritasi ini

diperolehi dengan menggunakan kaedah susuan kedua Rungge-Kutta(modified) Euler.

Perbincangan dan kesimpulan akan dibentangkan di dalam bab lima.

vii

TABLE OF CONTENTS

Page

SUPERVISOR’S DECLARATION ii

STUDENT’S DECLARATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF SYMBOLS xiii

LIST OF ABBREVIATIONS xv

CHAPTER 1 INTRODUCTION

1.1 Navier-Strokes Equation 1

1.2 Computational Fluid Dynamics 2

1.3 Lattice Boltzmann Method (LBM) 2

1.4 Classical CFD Versus LBM Method 4

1.5 Project Objective 5

1.6 Project Scopes 5

1.7 Project background 6

1.8 Thesis Outline 6

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction 8

2.2 Governing Equation 9

2.3 Basic Principle 10

viii

2.4 Collision Integral 12

2.5 Time Relaxation 13

2.6 Discretization of Microscopic Velocity 13

2.6.1 Isothermal Fluid Flow 14

2.6.1.1 The Macroscopic Equation For Isothermal 14

2.6.2 Thermal Fluid Flow 15

2.6.2.1 The Macroscopic equation For Thermal 15

2.7 Bhatnagar-Grook-Krook (BGK) 16

2.8 Boundary Condition 16

2.8.1 Bounce Back 17

2.8.1 Periodic Boundary 18

2.9 Theory of Poiseulle Flow 18

2.10 Theory of Couette Flow 19

2.11 Definition Of Rayleigh Number, Reynolds Number And Prandtl

Number

20

2.11.1 Rayleigh Number 20

2.11.2 Reynolds Number 21

2.11.3 Prandtl Number 21

CHAPTER 3 METHODOLOGY

3.1 Algorithm 23

3.2 Flow Chart For the Thesis 25

3.3 Simulation Result of Poiseulle Flow 27

3.4 Simulation Result of Couette Flow 28

3.5 Simulation Result of Porous Couette Flow 29

CHAPTER 4 RESULTS AND DISCUSSION

4.1 Finite Difference lattice Boltzmann Method 33

4.2 Finite Difference Thernal Lattice Boltzmann Method 34

4.3 Natural Convection in a Square Cavity 35

4.4 Expected result for the simulation of natural convection in a

square cavity at ra=105

38

ix

CHAPTER 5 CONCLUSION AND RECOMMENDATIONS

5.1 Conclusion 40

5.2 Recommendations 41

REFERENCES

42

APPENDICES

A1 Gannt chart for final year project 1 44

A2 Gannt chart for final year project 2 45

B Finite difference simulation 46

C Mesh for 2d natural convection 60

x

LIST OF TABLE

Table No. Title Page

4.1 Comparison among the present result with other LBM 39

xi

LIST OF FIGURES

Figure No. Title Page

1.1 Historically stages in the development of lattice Boltzmann model 4

1.2 Classical CFD versus LBM 5

2.1 General concept of Lattice Boltzmann 9

2.2 Streaming and collision processes 10

2.3 Time relaxation concept 13

2.4 Lattice Boltzmann Isothermal Model 14

2.5 Lattice Boltzmann Thermal Model 15

2.6 Schematic plot of bounce back boundary condition 17

2.7 Periodic boundary condition 18

3.1 Original LBM algorithm flowchart 24

3.2 Flow chart for the relation in the thesis 25

3.3 Poiseulle flow graph 28

3.4 Couette flow graph 29

3.5 Temperature profile at Ra=100 and Re=10 and Pr=0.2, 0.8, and

1.5

30

3.6 Temperature profile at Pr = 0.71 and Ra = 100 and Re=5, 10, 20,

and 30

31

3.7 Velocity and temperature profile at Re=10 , Pr = 0.71 and Ra =

60000

31

4.1 The schematic for natural convection in a square cavity 36

xii

4.2 Streamline and isotherms for the simulation of natural convection

in a square cavity for Ra=103

37



37



38

xiii

LIST OF SYMBOLS

R Gas constant

t Time

T Temperature

TC Cold temperature

TH Hot temperature

u Horizontal velocity

u Velocity vector

U Horizontal velocity of top plate

v Vertical velocity

V Volume

x Space vector

P Pressure

τ f ,g Time relaxation

υ Shear viscosity

β Thermal expansion coefficient

ε Internal energy

ρ Density

𝑇∞ Infinity temperature

𝑇𝑓 Film temperature

𝑇𝑠 Surface temperature

A Area of contact A

xiv

η Proportionally constant

χ Thermal diffusivity

Ω Collision operator

𝑇𝑚 Average temperature

g Acceleration due to gravity

c Microscopic velocity

𝑓𝑒𝑞 Equilibrium distribution function

f Distribution function

Pr Prandtl number

Ra Rayleigh number

Re Reynolds number

xv

LIST OF ABBREVIATIONS

BGK Bhatnagar Gross krook

CFD Computational fluid dynamics

LBM Lattice Boltzmann method

LGA Lattice gas approach

LGCA Lattice gas cellular automata

PBC Periodic Boundary condition

PDEs Partial differential equations

CHAPTER 1

INTRODUCTION

1.1 NAVIER-STROKES EQUATION

The Navier-strokes equation can derive as the motion of fluid substances that is

substances which can flow. These equations arise from applying Newton's second law to

fluid motion, together with the assumption that the fluid stress is the sum of a diffusing

viscous term (proportional to the gradient of velocity), plus a pressure term. The derivation

of the Navier–Stokes equations begins with the conservation of mass, momentum, and

energy being written for an arbitrary control volume. These equations describe how the

velocity, pressure, temperature, and density of a moving fluid are related. The mathematical

relationship governing fluid flow is the famous continuity equation and Navier-strokes

equation is given by

. 0 u (1.1)

22 1

.6

uP

t

u u u (1.2)

The Navier-Stroke equation is nonlinear partial differential equations in almost

every real situation and so complex that there is currently no analytical solution to them

except for a small number of special cases. The Navier–Stokes equations dictate not

position but rather velocity. A solution of the Navier–Stokes equations is called a velocity

2

field or flow field, which is a description of the velocity of the fluid at a given point in

space and time. Once the velocity field is solved for, other quantities of interest (such as

flow rate or drag force) may be found. (C.S. Nor Azwadi, 2007)

Nowadays, the use of a computer is necessary to determine the fluid motion of a

particular problem because fluid related problems arising in science and engineering are

extremely complex by nature.

1.2 COMPUTATIONAL FLUID DYNAMICS

Computational Fluid Dynamic (CFD) is a based on the fundamental governing

equation of fluid dynamics-the continuity, momentum and energy equations. The most

fundamental consideration in CFD is how one treats a continuous fluid in a discretized

fashion on a computer. One method is to discretize the spatial domain into small cells to

form a volume mesh or grid, and then apply a suitable algorithm to solve the Navier-

Strokes equation or an equation derived from them. To exactly simulate fluid flow in a

computer it would be necessary to solve Navier-Stroke equation with infinite accuracy .In

reality, numerical researchers must choose a method to discretize the problem. Some of the

general numerical methods used in computational fluid dynamics are described here.

1.3 LATTICE BOLTZMANN METHOD (LBM)

The Lattice Boltzmann method (LBM) is a recently developed method for

simulating fluid flows and modeling physics in fluids. Unlike the traditional CFD methods,

which solve the conservation equations of macroscopic properties (i.e., mass, momentum,

and energy) numerically, LBM models the fluid consisting of fictive particles, and such

particles perform consecutive propagation and collision processes over a discrete lattice

mesh. Due to its particulate nature and local dynamics, LBM has several advantages over

other conventional CFD methods, especially in dealing with complex boundaries,

incorporating of microscopic interactions, and parallelization of the algorithm. It is also

3

known as an alternative approach to the well-known finite difference, finite element and

finite volume technique for solving the Navier-Strokes equations. Lattice Boltzmann

methods (LBM) is a class of computational fluid dynamics (CFD) methods for fluid

simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann

equation is solved to simulate the flow of a Newtonian fluid with collision models such as

Bhatnagar-Gross-Krook (BGK).LB scheme is a scheme evolved from the improvement of

lattice gas automata and inherits some features from its precursor, the Lattice Gas Automata

(LGA).

The LBM recognizes that Boltzmann’s transport equation is a computational tool

that can be solved on the lattice .The collision term of this equation can be simplified using

the Bhatnagar-Gross-Krook (BGK) approximation where the distribution function relaxes

to a local equilibrium with a constant relaxation time. The main motivation for the

transition from LGA to LBM was the desire to remove the statistical noise by replacing the

Boolean particle number in a lattice direction with its ensemble average, the so-called

density distribution function. Accompanying this replacement, the discrete collision rule is

also replaced by a continuous function known as the collision operator. In the LBM

development, an important simplification is to approximate the collision operator with the

Bhatnagar-Gross-Krook (BGK) relaxation term. This lattice BGK (LBGK) model makes

simulations more efficient and allows flexibility of the transport coefficients. (Xiaoyi He

and Li-Shi Luo ,1997)

Although LBM approach treats gases and liquids as systems consisting of

individual particles, the primary goal is to build the connection between the microscopic

and macroscopic dynamics, rather than to deal with macroscopic dynamics directly. In

other words, the goal is to derive macroscopic equations from microscopic dynamics by

means of statistic, rather than to solve macroscopic equation.

Figure 1.1: Historically stages in the development of lattice Boltzmann model

Source: Wolf Gladrow, 2000

1.4 CLASSICAL CFD VERSUS LATTICE BOLTZMANN METHODS

The conventional simulation of fluid flow and other physical processes generally

starts from non linear partial differential equation (PDEs). These PDEs are discretized

by finite differences, finite element finite volume or spectral methods. The resulting

Lattice Boltzmann equation have been used at the

cradle of lattice gas cellular automata by Ferish etal

in 1987 to calculate the viscosity of LGCA.

McNamara and Zanetti introduced LBM as a

independent numerical method for the hydrodynamic

simulation in 1988.Fermi Dirac distributions were

used as equilibrium functions.

Higuera and Jimenez introduced linearized collision

operator in 1989.

Replacement of the Fermi Dirac distribution

functions with Boltzmann Maxwell distribution

function.

Linearized collision operator has benn replaced by

BGK (also called single time relaxation)

approximation by Koelman in 1991 and Qian et al. in

1992

5

algebraic equations of ordinary differential equation are solved by standard numerical

methods. In LBM, the starting point is a discrete microscopic model which by

construction conservation equation of mass and momentum for Navier-Stokes equation.

The derivation of the corresponding macroscopic equation requires multi-scale analysis.

(Wolf Gladrow, 2000)

Discretization

Figure 1.2: Classical CFD versus LBM

1.5 PROJECT OBJECTIVE

To develop finite difference Lattice Boltzmann Scheme for the Natural Convection

Heat Transfer

1.6 PROJECT SCOPES

The first project scope is to analysis heat transfer limit to natural convection

only. The second project scope is the problem will be test at Rayleigh number, Ra =103

to 105.This limitation due to Lattice Boltzmann Method (LBM) cam perform well at

low Rayleigh number and at high Rayleigh number LBM having a problem. This

selection high Rayleigh number is to show that this scheme can simulate problem at

high Rayleigh number. For the last project scope is to simulate natural convection in a

Partial Differential equation

(Navier-Strokes Equation)

Partial Differential equation

(Navier-Strokes Equation)

Ordinary differential Equation

(Solved using standard

numerical method)

Discrete Model (Lattice

Boltzmann Medels)

6

square cavity. Detail characteristic numerical value of the flow will be carrying out;

isotherm line, stream line and average of Nusselt number where they occur will be

compared.

1.7 PROJECT BACKGROUND

The lattice Boltzmann Method (LBM) is an alternative approach to the well-

known finite difference, finite element and finite volume techniques for solving the

Navier-Strokes equations. LB scheme is a scheme evolved from the improvement of

lattice gas automata (LGA) and inherits some features from its precursor, the LGA. The

implementation of the Bhatnagar-Gross-Krook (BGK) approximation is a improvement

to enhance the computational efficiency has been made for the LB method. The

algorithm is simple and can also easily modify to allow for the application of other,

more complex simulation component. In mathematics, finite difference methods are

numerical methods for approximating the solution to difference equation using finite

difference equation to approximate derivatives. Finite difference lattice Boltzmann

method is obtained using second order Runge-Kutta (modified) Euler Method.

1.8 THESIS OUTLINE

The aim of this thesis is to study the methods of the lattice Boltzmann equations

in order by using finite difference method. These subjects, newly emerged in 1980’s

utilize the statistical mechanics of simple discrete models to simulate complex physical

systems. The theory of lattice Boltzmann method in 4-discrete velocity and 9-discrete

velocity are reviewed in detail.

In the Chapter two, the concept of a distribution function is considered and the

derivation and the theory of the classical Boltzmann equation are discussed briefly.

Then the theory of lattice Boltzmann method and its coefficients from the Boltzmann

equation (Chapman-Enskog expansion) are also presented.

For the four discrete velocities is found in the isothermal model and for the 9-

discrete velocity, happen in thermal model that will discuss in chapter three. By using 4-

7

type of discrete velocity, we can apply to develop Porous Couette flow problem for the

thermal fluids problem. Using 9-discrete velocity model can apply in Poiseulle flow and

Couette flow

In Chapter four, the simulation by using Finite difference method was applied

among the first approaches applied to the numerical solution of differential equations.

This method is directly applied to the differential form of the governing equations. The

finite difference is a second order equation upwind scheme.

Finally in chapter five, conclusions and discussion on future studies are

presented and also some of recommendations will elaborate in order to improve the

project on future.

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

For the literature review will consist the theory of Lattice Boltzmann Model

(LBM) that a content are governing equation, basic principle of LBM, collide function

Bhatnagar-Gross-Krook (BGK), equilibrium distribution function, time relaxation,

discretization of microscopic velocity and lastly is a derivation of Navier-Strokes

equation. For the derivation of Navier-Strokes equation was already state in Chapter 1.

Beside, derivation of two type of boundary condition which are Bounce-back and also

periodic also presented. For this chapter also, we state briefing about Isothermal Lattice

Boltzmann Model which are Poiseulle Flow and Couette Flow and Thermal Lattice

Boltzmann Model which is Porous Couette Flow.

LBM is a computational fluid dynamics (CFD) method or alternative method to

simulate the fluid problems especially to simulate the complex fluid flow problems

including single and multiphase flow in complex geometries. The main objective to

achieve is to create a connection between the microscopic and macroscopic dynamics.

From the Navier-Strokes equation, we already know the density, velocity, pressure and

etc. So, this is called macroscopic scale. But, for the Boltzmann equation, we only have

the particles that moves to one place to another and called microscopic scale

9

Figure 2.1: General concept of Lattice Boltzmann

Source: C.S. Nor Azwadi, 2007

2.2 GOVERNING EQUATION

From this equation, we can the relation between microscopic and macroscopic

dynamics. The Boltzmann equation given is shown below:-

( , ) ( , ) ( )f x c t t t f x t f (2.1)

Where f = density distribution function

c = microscopic velocity

Ω = collision integral

Moment of distribution function

Macroscopic variables

Density,velocity, pressure, etc.

Statistical Mechanics

Density distribution function, ( , )f x t

10

2.3 BASIC PRINCIPLE

Basic principle of LBM is including streaming step and collision step. The

particles move to another place in the variable direction with their velocities (streaming

step) and after they meet to each other, the collision happens (collision step) and the

particles will separate again. (Streaming step).Example is from the ‘snooker’. From this

situation, means when a ball hit to another ball, its can firstly streaming and then it will

collision and it become streaming again.

Figure 2.2: Streaming and collision processes


It represents the minimal from of Boltzmann kinetic equation (Higuera and

Jimenez, 1989) and the result is a very elegant and simple equation. (F.Kuznik et al.

2007). Lattice Boltzmann equation is directly obtained from the lattice gas automata by

taking ensemble average with the assumption of random phase and leads to the

following equation. (Wolf Gladrow, 2000)

( , , ) ( , , ) ( )f x c t c a t t t f x c t f (2.2)

Where ( , , )f x c t = the single particle distribution function with discrete velocity, c

( )f = the lattice Boltzmann collision operator

Streaming

Streaming

Collision

11

Distribution function 𝑓 𝑥, 𝑐, 𝑡 describe the number of particles at position

𝑥,moving with velocity, 𝑐 at time, 𝑡. There are two conditions related to the distribution

function; without collisions and with collisions. At a short time,∆𝑡 each particle would

move from 𝑥 to 𝑥 + 𝑐∆𝑡 and each particle velocity would change from 𝑐 to 𝑐 + 𝑎∆𝑡

where 𝑎 is the acceleration due to external forces on a particle at 𝑥 with a velocity𝑥.

Hence, the number of molecules 𝑓 𝑥, 𝑐, 𝑡 𝑑𝑥𝑑𝑐 is equal to the number of molecules

𝑓 𝑥 + 𝑐∆𝑡, 𝑐 + 𝑎∆𝑡, 𝑡 + ∆𝑡 𝑑𝑥𝑑𝑐 for the distribution without collisions. (N. A. C.

Sidik, 2007). Therefore;

( , , ) ( , , ) 0f x c t c a t t t dxdc f x c t (2.3)

There will be a net difference between the number of molecules 𝑓 𝑥, 𝑐, 𝑡 𝑑𝑥𝑑𝑐

and the number of molecules 𝑓 𝑥 + 𝑐∆𝑡, 𝑐 + 𝑎∆𝑡, 𝑡 + ∆𝑡 𝑑𝑥𝑑𝑐 if collision occurs

between the molecules.

This can be expressed by;

( , , ) ( , , ) ( )f x c t c a t t t dxdc f x c t dxdc f (2.4)

Where Ω(𝑓)𝑑𝑥𝑑𝑐𝑑𝑡 is the collision operator. On dividing by 𝑑𝑐𝑑𝑡 , and letting

𝑑𝑡 tends to zero 𝑑𝑡~ 0 give the Boltzmann equation for 𝑓 ;

( )a

a a

f f fc a f

t c c

(2.5)

Without collision;

𝑓 𝑥, 𝑐, 𝑡 𝑓 𝑥 + 𝑐∆𝑡, 𝑐 + 𝑎∆𝑡, 𝑡 + ∆𝑡

At time, 𝑡 at time, ∆𝑡

12

With collision;

𝑓 𝑥, 𝑐, 𝑡 𝑓 𝑥 + 𝑐∆𝑡, 𝑐 + 𝑎∆𝑡, 𝑡 + ∆𝑡

Ω(𝑓)

At time, 𝑡 at time, ∆𝑡

2.4 COLLISION INTEGRAL, Ω

The BGK collision model assumes that the system is at near equilibrium state

and the particle distribution function relaxes to its equilibrium state at a constant rate. It

term refers to a collision operator used in the Boltzmann Equation and in the Lattice

Boltzmann method, a Computational fluid dynamics technique. Boltzmann’s equation

describes the evolution of molecules in rare gas. If no external force is present, then

after a long time the gas should reach an equilibrium state. Boltzmann’s equation

describes such behavior. Boltzmann conceived the H-theorem to explain how a many-

body system to approach equilibrium from an arbitrary non-equilibrium initial state. He

derived this theorem based on the assumption of ‘molecular chaos’ where the two

colliding molecule’s position and velocities were statistically uncorrelated or unrelated.

(S. Succi, 2001)

1

( ) ( )eqf f f

(2.6)

Boltzmann came out with the H-theorem where the value of distribution

function will always tend to the equilibrium distribution function, feq

during collision

process. The distribution function f can be related to feq

and τ is a relaxation parameter.

13

2.5 TIME RELAXATION

To achieve the equilibrium, means τ is a relaxation parameter (time to reach

equilibrium state during every collision process) and value of the time relaxation is

between (0.5< τ <1). Below shown the time relaxation concept:

τ = 0.5

τ = 0.6

τ=0.8

τ=0.9

τ=1.0

Non-equilibrium Equilibrium Non-equilibrium

Figure 2.3: Time relaxation concept

The figure above show value of τ = 0.5 is the limit for the relaxation time. The

value of time relaxation 𝜏 need to be more close to 1. The more close time relaxation 𝜏

to 1, the more number of particles exchange to equilibrium state.

2.6 DISCRETIZATION OF MICROSCOPIC VELOCITY

For the discretization of microscopic velocity, from the Gauss-Hermitte

integration, we can integrate from the continuous velocity to the 9-discrete velocity and

also 4-discrete velocity. We focused on the two type of discrete velocity. Means that for

the 9-discrete velocity happen in isothermal fluid flow (poiseulle flow and couette flow)

while the 4-discrete velocity happen in thermal fluid flow (porous couette flow).

14

2.6.1. Isothermal Fluid Flow

Figure below (Figure 2.4) shows the Lattice Boltzmann Isothermal Model

(Poiseulle Flow and Couette flow)

Figure 2.4: Lattice Boltzmann Isothermal Model


2.6.1.1 The Macroscopic Equation for Isothermal

By using chapmann-enskog expansion procedure, we can have the navier-stroke

equations accurate in continuity equations and in momentum equation:

∇. 𝒖 = 0 (2.7)

22 1.

6

uP

t

u u u (2.8)

12

3

4

5 98

7 6

Continuous velocity

Gauss-Hermitte integration

15

The relation between the time relaxation τ, in microscopic level and viscocity of

fluid ν, in macroscopic level is;

2 1

6

(2.9)

2.6.2 Thermal Fluid Flow

Figure 2.5 is shown the Lattice Boltzmann Thermal Model means Porous

Couette Flow

Figure 2.5: Lattice Boltzmann Thermal Model


2.6.2.1. The Macroscopic Equation for Thermal

By using the chapmann-enskog expansion procedure, we can get the derivation

equation for the energy equation.

21.( ) ( )

2g

TT T

t

u (2.10)

Gauss-Hermitte integration

Continuous velocity

16

Where:

1

32

f (2.11)

1

2g (2.12)

2.7 BHATNAGAR-GROSS-KROOK (BGK)

BGK is a combination step between streaming process and collision process.

These two processes are repeating one after another until all the distribution function

relaxes to the equilibrium distribution function. (S. Succi, 2001)

( , ) ( , )eqf f

f x c t t t f x t

(2.13)

Streaming process Collision process

fi: density distribution function

τ: relaxation parameter

feq

: equilibrium distribution

2.8 BOUNDARY CONDITION

The boundary conditions are responsible to determining these unknown

distributions. Basically, there are two ways to define boundary conditions; placing the

boundary on grid modes or placing the boundary on links (Xiaoyi he et al. 1995).

Lattice Boltzmann method have several types of boundary conditions.

17

2.8.1 Bounce-Back

The initial approach to simulate boundary was to follow the methods used in the

lattice Gas Approach. The simplest boundary condition is called bounce back boundary

conditions where all the distribution functions at the boundaries back along to the link

they arrived. The bounce-back boundary condition for lattice Boltzmann simulations is

evaluated for flow about an infinite periodic array of cylinders. The solution is

compared with results from a more accurate boundary condition formulation for the

lattice Boltzmann method and with finite difference solutions. The bounce-back

boundary condition is used to simulate boundaries of cylinders with both circular and

octagonal cross-sections. Figure 2.6 below show the schematic plot of the bounce-back

boundary condition. (Gallivan, Martha A., Noble, David R, Georgiadis, John G,

Buckius, Richard O, 1997)

Figure 2.6: Schematic plot of bounce back boundary condition

Source: Frisch et al., 1986

http://www.sandia.gov/eesector/gs/gc/hws/saltfing.htm#Frisch_etal_1986

18

2.8.2 Periodic Boundary

Periodic boundary conditions typically intended to isolate bulk phenomena from

the actual boundaries of the real physical system and consequently they are adequate for

physical phenomena where surface effect play a negligible role. Periodic boundary

conditions are applied directly to the particle populations, and not to macroscopic flow

variables. They are generally useful for capturing flow invariance in a given direction. If

a uniform body force is used instead of an imposed pressure gradient, periodic

conditions can be used in place of macroscopic inflow/outflow conditions in the stream

wise direction [Robert S. Maier et al.1996]. This boundary condition can be

implementing by bring the same distribution function that leaving the outlet to the inlet.

This can review with reference to Figure 2.7 that show a particles flow.

Rigid body

Periodic BC Fluid flow Periodic BC

Figure 2.7: Periodic boundary condition

2.9 THEORY OF POISEULLE FLOW

The Hagen-Poiseuille equation is a physical law that describes slow viscous

incompressible flow through a constant circular cross-section. It is also known as the

Hagen-Poiseuille law, Poiseuille law and Poiseuille equation. The Hagen Poiseuille

equation can be derived from the Navier-Stokes equations. The derivation of

Poiseuille's Law is surprisingly simple, but it requires an understanding of viscosity.

When two layers of liquid in contact with each other move at different speeds, there will

19

be a force between them. (S. P. Sutera, R. Skalak,1993). This force is proportional to

the area of contact A, the velocity difference in the direction of flow Δvx/Δy, and a

proportionality constant η and is given by

cos ,

xvis ity topF A

y

(2.14)

The negative sign is in there because we are concerned with the faster moving

liquid (top in figure).

2.10 THEORY OF COUETTE FLOW

In fluid dynamics, Couette flow refers to the laminar flow of a viscous fluid in

the space between two parallel plates, one of which is moving relative to the other. The

flow is driven by virtue of viscous drag force acting on the fluid and the applied

pressure gradient parallel to the plates.

Couette flow is frequently used in undergraduate physics and engineering

courses to illustrate shear-driven fluid motion. The simplest conceptual configuration

finds two infinite, parallel plates separated by a distance h. One plate, say the top one,

translates with a constant velocity u0 in its own plane. Neglecting pressure gradients, the

Navier-Stokes equations simplify to

2

20

d u

dy (2.15)

Where y is a spatial coordinate normal to the plates and u (y) is the velocity

distribution. This equation reflects the assumption that the flow is uni-directional. That

is only one of the three velocity components (u, v, w) is non-trivial. If y originates at the

lower plate, the boundary conditions are u (0) = 0 and u (h) = u0. The exact solution

found by integrating twice and solving for the constants using the boundary condition.

(B.R. Munson, D.F. Young, and T.H. Okiishi, 2002)

20

0( )y

u y uh

(2.16)

2.11 DEFINITION OF RAYLEIGH NUMBER, REYNOLDS NUMBER AND

PRANDTL NUMBER

2.11.1 Rayleigh Number

The Rayleigh number for a fluid is a dimensionless number associated with buoyancy

driven flow (also known as free convection or natural convection). When the Rayleigh

number is below the critical value for that fluid, heat transfer is primarily in the form of

conduction; when it exceeds the critical value, heat transfer is primarily in the form of

convection. The Rayleigh number is describes the relationship between buoyancy and

viscosity within a fluid and the Prandtl number which describes the relationship

between momentum diffusivity and thermal diffusivity.

For free convection near a vertical wall, this number is

3Pr Pr ( )x x s

gRa Gr T T x

(2.17)

In the above, the fluid properties Pr,𝜈, 𝛼 and 𝛽 are evaluated at the film temperature,

which is defined as

2

sf

T TT

(2.18)

21

2.11.2 Reynolds Number

Reynolds number can be defined for a number of different situations where a

fluid is in relative motion to a surface. These definitions generally include the fluid

properties of density and viscosity, plus a velocity and a characteristic length or

characteristic dimension.

For flow in a pipe or a sphere moving in a fluid the diameter is generally used

today. The velocity may also be a matter of convention in some circumstances, notably

stirred vessels.

ReVD VD QD

A

(2.19)

2.11.3 Prandtl Number

The Prandtl number is a dimensionless number approximating the ratio of

momentum diffusivity (kinematics viscosity) and thermal diffusivity. It is defined as:

PrpC

k

(2.20)

Typical values for Pr are:

i. Around 0.7-0.8 for air and many other gases

ii. Around 0.16-0.7 for mixture of noble gases or noble gases with hydrogen

iii. Around 7 for water

iv. Between 100 and 40000 for engine oil

v. Between 4 and 5 for R-12 refrigerant

vi. Around 0.015 for mercury

22

Prandtl number is related to the thickness of the momentum and thermal

boundary layer. When Pr is small, means that the heat diffuses very quickly compared

to the velocity (momentum). This means that for liquid metals the thickness of the

thermal boundary layer is much bigger than the velocity boundary layer.

CHAPTER 3

METHODOLOGY

For this chapter, we will discuss about the simulation result of poiseulle flow,

Couette flow and Porous Couette Flow. We also discuss about finite difference method

to simulate natural convection in a square cavity.

3.1 ALGORITHM

The algorithm flowchart for LBM is shown in Figure 3.1. It consists of two

processes; advection process and collision process. The initial values of density

distribution 𝑓 are specified at each grid point. Then, the system evolves in the following

steps.

i. The advection term is solved by applying the streaming process of the density

distribution function.

ii. Then the collision process is solved by BGK collision model.

iii. Next step is to define the boundary conditions based on the bounce back

boundary conditions.

iv. The convergence criterion is based on steady state conditions and will discuss in

Chapter 4.

24

Figure 3.1: Original LBM algorithm flowchart

MAIN PROGRAM

INITIAL CONDITION

ADVECTION

COLLISION

BOUNDARY CONDITION

OUTPUT

END CONVERGE

ITERATION

25

3.2 FLOW CHART FOR THE THESIS

Theory of lattice Boltzmann

i. Governing equation

ii. Basic principle

iii. Collision integral

iv. Time relaxation

v. Bhatnagar-Gross-Krook (BGK)

vi. Discretization of microscopic velocity

vii. Derivation of Navier-stroke equation

Isothermal fluid flow

i. Simulate Poiseulle flow

ii. Simulate Coutte flow

Extension to thermal LBM

i. Theory of thermal LB model

ii. Simulate Porous Coutte flow

Theory of finite difference

26

NO YES

Figure 3.2: Flow chart for the relation in the thesis

Theory of finite difference

Simulate the natural convection in

a square cavity by using Fortran 90

Check value of Nusselt

number with previous study

Graphical the result by using

AVS/Express Visualization

Edition

End

Change value of

Δt from 0.01 to

0.001, 0.0001

27

3.3 SIMULATION RESULT OF POISEULLE FLOW

Numerical simulation for the Poiseulle flow driven by a pressure gradient was

carried out to test the validity of the isothermal lattice Boltzmann model. In this case the

pressure gradient is set between the inlet and outlet end of the channel. This is done by

setting the density (proportional to the pressure) at slightly different values between the

two ends. The velocity is not set to any value. The well-known bounce-back boundary

condition is applied at the top and bottom wall.

Letting the system evolved, it is observed that it reaches a steady state

corresponding to the parabolic solution of the channel flow. The criterion of steady state

is set by

, 1 , 2i ii x

i

f x t f x t

i M N

(3.1)

Where M and N are mesh number in x and y direction respectively. It usually

takes a few thousand iterations to reach a steady state depending on the value of the

viscosity and the boundary conditions. Two type of measurement were taken in the

simulation. One is the measurement of velocity, u .The other is the measurement of

pressure along the channel. All the measurements were taken after the steady state is

attained. (C.S. Nor Azwadi, 2007)

28

Figure 3.3: Poiseulle flow graph

The density change between the two ends along the centerline of the channel is a

straight line as shown in Figure 3.3 .In the inset the velocity profile across the channel is

displayed for the stationary state. The figure corresponds to a simulation using lattice

size (4x33) means XD = 4, YD = 33 (corresponding to L = 30) and τf = 0.55

The combination of results shows that, not only the velocity profile is correct

(parabolic shape) but also the pressure distribution is linear along the channel length.

3.4 SIMULATION RESULT OF COUETTE FLOW

For this section, a numerical experiment involving the time evolution of the

Couette flow is presented, in which the top plate moves with constant velocity, while

the bottom plate is held fixed. The initial conditions correspond to a null velocity

everywhere expect on the top boundary, where the velocity is u = (1, 0). The x-

component of the velocity on the top plate is maintained at U=1.00 (top plate boundary

condition in LBM units), whereas the bottom one is at rest. No pressure gradient is

included for this case. The channel is L= 40.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Poiseulle flow

29

Figure 3.4: Couette flow graph

Figure 3.4 shows a sequence of normalized velocity profiles for different times.

The lattice size for this experiment is (4x32) where are XD = 4, YD = 32 and the

relaxation time is τf = 1.000. The velocity profiles are drawn at times t= 200, 445, 600,

1000, 1500 in LBM units. Periodic boundary conditions are implemented in the x-

direction. The solution for the steady state case well known and corresponds to the

velocity increasing linearly from zero at the bottom to U at the top plate.

3.5 SIMULATION RESULT OF POROUS COUETTE FLOW

For this section, we shall apply the newly developed model to simulate heat

transfer in porous coquette flow problem.

Consider two infinite parallel plates separated by a distance of L. The upper cool

plate at temperature TC moves at speed U and the lower hot plate at temperature TH is

stationary. A constant normal flow of fluid is injected through the bottom hot plate and

withdrawn at the same rate from the upper plate. The analytical solution of the velocity

field in the steady state is given by

00.10.20.30.40.50.60.70.80.9

1

0 0.2 0.4 0.6 0.8 1

YD

XD

Coutte flow

200

445

600

1000

1500

30

Re

Re

1( )

1

y

Leu U

e

(3.2)

Where Re is the Reynolds number based on the inject velocity,vo. The

temperature profile in the steady state satisfies

Pr Pr Re

Pr Pr Re

1Δ

1

y

L

c

eT T T

e

(3.3)

Where ΔT=TH-Tc is the temperature difference between the hot and cool walls.

Pr=v/χ is the Prandtl number. Another dimensionless parameter id the Rayleigh number

defined by

3Δg TL

Ra

(3.4)

Figure 3.5: Temperature profile at Ra=100 and Re=10 and Pr=0.2, 0.8, and 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

0.2

0.8

1.5

31

Figure 3.6: Temperature profile at Pr = 0.71 and Ra = 100 and Re=5, 10, 20, and 30

Figure 3.7: Velocity and temperature profile at Re=10, Pr = 0.71 and Ra = 60000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

10

5

20

30

40

32

Periodic boundary conditions are used at the entrance and exit of the channel

and the non-equilibrium bounce back boundary conditions for the velocity. The

normalized temperature profile for Ra=100, Re=10 and Pr =0.2, 0.8, and 1.5 are shown

in figure 3.5. For the figure 3.6 shows the result for the Pr=0.71, Ra= 100 and Re= 5,

10, 20 and 30. They agree well with the analytical solution. To show that this model is

suitable and numerically stable for a wide range of Rayleigh number, the computations

for Ra=10 till Ra=60000 at Pr=0.71 and Re=10 have been done. The results are shown

in figure 3.7. The results presented above indicating that the numerical stability of the

new model is higher than the previous proposed 4-velocity lattice model.

33

CHAPTER 4

RESULT AND DISCUSSION

4.1 FINITE DIFFERENCE LATTICE BOLTZMANN METHOD

One possible way to release the constraint of the lattice symmetry is to use the

fnite difference scheme for the lattice Boltzmann equation. For this section, the

discretization procedure of LBE is demonstrated with finite difference technique. The

temporal discretization is obtained using second order Rungge-Kutta (modified) Euler

method. The time evolution of particle distributions is then derived by

1 ,

2 .2

n eq nn

n n i ii i i i

f

f ftf f c f

(4.2)

1 1,

1 2 21 2.

n eq nn

n n i ii i i i

f

f ff f t c f

(4.3)

The second order Equations of 4.4~4.7 upwind schemes can be applied to

calculate the spatial gradient in equation 4.1;

1

. , ( )eqii i i i

f

fc f x t f f

t

(4.1)

3 , 4 , ( 2 , )

, 02

i i i

ix x i ix ix

f x y f x x y f x x yc f c c

x

(4.4)

34

3 , 4 , ( 2 , )

, 02

i i i

ix x i ix ix

f x y f x x y f x x yc f c c

x

(4.5)

3 , 4 , ( , 2 )

, 02

i i i

iy y i iy iy

f x y f x y y f x y yc f c c

y

(4.6)

3 , 4 , ( , 2 )

, 02

i i i

iy y i ix iy

f x y f x y y f x y yc f c c

y

(4.7)

The combination of these specifics space and time discretization results in

second order in space and second order in time. (C.S.Nor Azwadi)

4.2 FINITE DIFFERENCE THERMAL LATTICE BOLTZMANN METHOD

The evolution of internal energy density distribution is given by;

1

. ( )eqii i i i

g

gc g g g

t

(4.8)

The time space can also be discretised using second order Rungge-Kutta

(modified) Euler method. The time evolution of particle distributions is then given by;

1 ,

2 .2 i

n eq nn

n n i ii i i g

g

g gtg g c

(4.9)

1 1,

1 2 21 2.

i

n eq nn

n n i ii i i g

g

g gg g t c

(4.10)

35

4.3 NATURAL CONVECTION IN A SQUARE CAVITY

The natural convection in a square cavity is brought to test the validity of finite

difference thermal lattice Boltzmann model using the newly developed 4-velocity model

for the internal energy density equilibrium density distribution function. The problem is

a two dimensional square cavity with side wall is a difference temperature which are a

hot wall on the left and the cold wall on the right side. Meanwhile, the top side and the

bottom side walls being adiabatic. The temperature difference between walls show a

temperature gradient in a fluid and also consequently density difference induces a fluids

motion. This called convection.

In this simulation, the Boussinesq approximation is applied to the buoyancy

force term. With this approximation, it assumed that all fluid properties β and υ can be

considered as constant in the body force term except for the temperature dependence of

the density in the gravity term.

( )mG g T T j (4.11)

Where β is the thermal expansion coefficient, g is the acceleration due to

gravity, 𝑇𝑚 is the average temperature and j is the vertical direction opposite to that of

gravity. The dynamical similarity depends on two dimensionless parameters; the prandtl

number, Pr and the Rayleigh number, Ra defined by

Pr

(4.12)

3Δg TL

Rax

(4.13)

36

Figure 4.1: The schematic for natural convection in a square cavity.


The simulation were done for Rayleigh number from Ra=103 to Ra=10

5.The

mesh size of 51x51, 101x101, 201x201 are used for Ra=103, Ra=10

4 and Ra=10

5

respectively. In this simulation, Prandlt number, Pr is set to be 0.71.The benchmark

result of the natural convection in a square cavity was based on the solution of Navier

Strokes equation by Davis (1983). Figure below show the streamlines and isotherms for

the simulation at Rayleigh number, Ra=103 to Ra=10

5

37

Figure 4.2: Streamline and isotherms for the simulation of natural convection in a

square cavity for Ra=103

\

At the beginning of the simulation for Ra= 103, vortex appear at the center of the

cavity with circular shape. The isotherms are almost vertically parallel to the wall

indicating that conduction mode heat transfer mechanism is dominant. From figure 4.2,

streamline and isotherms can see clearly.



38

For Ra= 104 from figure 4.3, circular vortex at the center of cavity was distorted

and its shape change to horizontal oval due to the connection effect. Meanwhile,

isotherms start to be horizontally parallel to the wall at the cavity center. This effect due

to the heat transfer mechanisms are mixed conduction and convection.

4.4 EXPECTED RESULT FOR THE SIMULATION OF NATURAL

CONVECTION IN A SQUARE CAVITY AT RA=105

Figure 4.4 show that the streamline and isotherms for the simulation at

Ra=10000 that have done by Azwadi. It is show that the increasing the Rayleigh

number to Ra=105, a vortex oval shape appear at the left of the cavity when the system

achieve equilibrium condition. All isotherms are almost horizontally parallel to the wall

indicating that the convection is the main heat transfer mechanism.



39

Table shows the comparison the average Nusselt number throughout the cavity

by using finite difference scheme for Ra=103 to Ra=10

6

Table 4.1: Comparison among the present result with other LBM.

Rayleigh number, ra 103 10

4 10

5 10

6

X.He et al (1998) 1.117 2.244 4.520 -

Peng et al.(2003) 1.117 2.235 4.511 -

Nuave Davis (1883) 1.116 2.234 4.510 8.798

Azwadi (2007) 1.117 2.236 4.549 8.723

Rosdzimin (2008) 1.116 2.201 4.249 -

Present (2009) 1.116 1.737 - -

From the table 4.1 shows the predicted results are compared with the results that

obtained by original double distribution function thermal lattice Boltzmann scheme

( X.He et al 1998), the simplified scheme (Peng et al. 2003), the solution by Navier-

Strokes equation ( Davis 1983), the simplified thermal (Azwadi 2007) and lastly by

using CIP-LBM scheme by (Rosdzimin 2008).

For all the value of Rayleigh number have been considered in the present

analysis where the average Nusselt number have been predicted ±0.5 error compare to

the previous result and can be accepted for the real engineering applications.

The evolution of lattice Boltzmann equations have been discretised using second

order upwind finite difference. From that equation, as expected that the boundary layer

is thicker than the velocity boundary layer for different Rayleigh number simulations.

The flow patterns that including the boundary layers and vortices can be seen clearly in

Figure 4.2, Figure 4.3 and also in Figure 4.4. The result obtained demonstrate that by

using double distribution function thermal lattice Boltzmann model is very efficient

procedure to study flow and heat transfer in a differentially heated square enclosure.

40

CHAPTER 5

CONCLUSION AND RECOMMENDATIONS

5.1 CONCLUSION

In chapter one, have been introduced about the Navier-stokes equation and

introduced the lattice Boltzmann method (LBM). In literature review, the theory of

lattice Boltzmann method (LBM) from the Boltzmann equation has been discussed. The

theory of the classical Boltzmann equation is also discussed.

In methodology, the algorithm of the advection and collision process was

explained detail in chapter three. Simulation result for the isothermal flow which are

Poiseulle flow and Couette flow have been discuss. Beside, the simulation result for the

thermal flow which is Porous Couette flow also has been performed well.

Chapter four concerned with the combination of the finite difference scheme

with the lattice Boltzmann method. Results for all the above fluid flow problems show

that LBM is a reliable CFD technique and agreed with the analytical solution and

conventional approach.

Objective for this thesis was achieved and already discussed in chapter four.

Simulation of natural convection in a square cavity by using finite difference method is

successfully done for the Ra=103, Ra=10

4. In this thesis, the advection term in both

density and internal energy density equations has been discretised by using second order

upwind scheme and applied in the simulation of natural convection in a square cavity.

41

By using finite difference scheme, the thickness of thermal boundary layer decrease as

Rayleigh number increase.

For the simulation at Rayleigh number, Ra=105 occur some problem. Simulation

at high Rayleigh number will take a longer time to simulate. In this case, Ra=105 need a

longer time, more than one month to simulate. Because of lack of time, the simulation is

not done yet. To increase the simulation time, we were forced to apply small value of

∆t. This problem proposed to be solved in recommendation for future work.

5.2 RECOMMENDATIONS

Modification of finite difference lattice Boltzmann scheme can be done by

solving the non advection terms using higher order Rungge-Kutta method in order to

reduce the simulation time by increasing the accuracy of time. Using this advantage,

finite difference lattice Boltzmann scheme can be extended for simulation of any fluid

flows problem or heat transfer problem using the non uniform grid size.

42

REFERENCES

B.R. Munson, D.F. Young, and T.H. Okiishi (2002). Fundamentals of Fluid Mechanics,

John Wiley and Sons. ISBN 0-471-44250-X

C. S. N. Azwadi and T. Tanahashi (2006). Simplified Thermal Lattice Boltzmann in

Incompressible Limit. International Journal of Modern Physics B. Vol. 20, No.

17 pp 2437-2449

C. S. N. Azwadi and T. Tanahashi (2007). Three-Dimensional Thermal Lattice

Boltzmann Simulation of Natural Convection in a Cubic Cavity. International

Journal of Modern Physics B. Vol. 21, No. 1 pp 87-96

C. S. N. Azwadi and T. Tanahashi (2008). Simplified Finite Difference Thermal Lattice

Boltzmann Method. International Journal of Modern Physics B. Vol. 22, No.22

pp 3865-3876

D. V. Davis (1983). Natural Convection of air in a square cavity: A benchmark

numerical solution. International Journal Numerical Method Fluid, 3, pp249-

26472

Gallivan, Martha A., Noble,David R, Georgiadis, John G, Buckius, Richard O (1997),

An Evaluation of the Bounce-Back Boundary Condition for Lattice Boltzmann

Simulations. International Journal of Numerical Methods in Fluids, vol. 25, Issue

3, pp.249-263

Dieter A. Wolf-Gladrow (2000). Lattice Gas Cellular Automata and Lattice Boltzmann

Models, An Introduction. Springer

Dieter Wolf-Gladrow 1 (1995). A Lattice Boltzmann Equation for Diffusion. Journal

of Statistical Physics, Vol. 79, Nos. 5/6, 1995

S. P. Sutera, R. Skalak (1993). The history of Poiseuille's law. Annual Review of Fluid

Mechanics, Vol. 25, pp. 1-19

Succi, Sauro (2001), The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.

Oxford University

Xiaoyi He and Li-Shi Luo (1997). A priori derivation of the lattice Boltzmann

equation. Physical Review E, volume 55, number 6

43

Xiaoyi He, Qisu Zou, Li-Shi Luo, and Micah Dembo (1997). Analytic Solutions of

Simple Flows and Analysis of Nonslip Boundary Conditions for the Lattice

Boltzmann BGK Model. Journal of Statistical Physics, Vol. 87, Nos. ½

Xiaoyi He, Shiyi Chen, and Gary D. Doolen (1998) A Novel Thermal Model for the

Lattice Boltzmann Method in Incompressible Limit. Journal Of Computational Physics

146, 282–300. Article No. Cp986057

44

APPENDIX A1

GANNT CHART FOR FINAL YEAR PROJECT 1

PROJECT

ACTIVITIES

W

2

W

3

W

4

W

5

W

6

W

7

W

8

W

9

W

10

W

11

W

12

W

13

W

14

W

15

Literature

Study

Lattice

Boltzmann

Isothermal

Flow

1. Poiseulle

Flow

2. Couette

Flow

Thermal Fluid

Flow

Porous

Couette Flow

Submit Report

Presentation

45

APPENDIX A2

GANNT CHART FOR FINAL YEAR PROJECT 2

PROJECT

ACTIVITIES

W

2 W

3 W4 W5 W6 W7 W8 W9 W10 W 11 W12 W13 W14 W15

Theory of finite

difference

method

Simulate the

natural

convection in a

square cavity for

1.Ra=103

2=Ra=104

3=Ra=105

Graphical the

result by using

AVS

Presentation

Submit Report

46

APPENDIX B

FINITE DIFFERENCE SIMULATION

!***********************************************************!

Finite Difference lattice Boltzmann

!***********************************************************!

program cavity

implicit real*8 (a-h,o-z)

parameter (ij = 330, kkk = 8)

common/var1/f(0:ij,0:ij,0:kkk),feq(0:ij,0:ij,0:kkk),g(0:ij,0:ij,0:k

kk),geq(0:ij,0:ij,0:kkk)

common /var2/ cx(0:kkk),cy(0:kkk),dx(0:kkk),dy(0:kkk)

common /var3/ u(ij,ij),v(ij,ij),temp(ij,ij)

common /var4/ rho(ij,ij)

common /var5/ tmp(0:ij,0:ij,0:kkk),ff(0:ij,0:ij,0:kkk)

common/var6/fn(0:ij,0:ij,0:kkk),fx(0:ij,0:ij,0:kkk),fy(0:ij,0:ij,0:

kkk),fxn(0:ij,0:ij,0:kkk),fyn(0:ij,0:ij,0:kkk)

common/var7/gn(0:ij,0:ij,0:kkk),gx(0:ij,0:ij,0:kkk),gy(0:ij,0:ij,0:

kkk),gxn(0:ij,0:ij,0:kkk),gyn(0:ij,0:ij,0:kkk)

common /con1/ delt,ra, pr, th ,tc,gra

common /con2/ lx,ly

common /con3/ ntin,nstep,totalg

common/con4/pi,tauf,rhoint,delx,dely,delx2,dely2,rtauf,xnyu,chi,tau

g,rtaug

integer lx,ly,ntin,nstep,i,j,k,l

integer ::unit, ierror

character (len=6)::filename

real vel(ij,ij)

lx = 51

ly = 51

ra = 1000.d0

pr = 0.71d0

th = 1.0d0

tc = 0.0d0

ntin = 10

delt = 0.001d0

rhoint = 1.0d0

uinit = 0.0d0

write(*,*)'lx',lx

write(*,*)'ly',ly

write(*,*)'ra',ra

write(*,*)'pr',pr

write(*,*)'th',th

write(*,*)'tc',tc

write(*,*)'ntin',ntin

write(*,*)'delt',delt

write(*,*)'rhoint',rhoint

write(*,*)'uinit',uinit

gra = (0.0557**2)/(lx-1)

xnyu = (gra*((lx-1)**3)*(th-tc)*pr/ra)**0.5

47

tauf = 3*xnyu

chi = xnyu/pr

taug = chi

rtauf = 1.0/tauf

rtaug = 1.0/taug

write (6,*) 'tauf = ',tauf

write (6,*) 'rtauf = ',rtauf

write (6,*) 'taug = ',taug

write (6,*) 'rtaug = ',rtaug

write (6,*) 'gra = ',gra

write (6,*) 'is everything ok?'

read (*,*) ok

pi = atan(1.0d0)*4.0d0

write (6,*) 'calculation start'

call initial

do nstep = 1, 5000000

call fin

call output

call equilibrium

if (mod(nstep,ntin) .eq. 0) then

totalg=0.0

do i = 1,lx

do j = 1,ly

do k=1,4

totalg=totalg+g(i,j,k)

end do

end do

end do

write(*,*) nstep

write(*,10)totalgn-totalg10

format ('Convergence= ',F10.7)

end if

if (mod(nstep,ntin) .eq.1) then

totalgn=0.0

do i = 1,lx

do j = 1,ly

do k = 1,4

totalgn=totalgn +g(i,j,k)

end do

end do

end do

end if

!***** if converge*****::!

if (abs(totalgn - totalg) .le. 1.0e-5 ) then

go to 100

end if

48

end do

100 write(*,*)'end of iteration'

open(unit=30,file='uvel1.dat',status='replace',action='write',iosta

t=ierror)

write(30,*)' Thermal Diffusivity ',chi

write(30,*)' Rayleigh Number ',ra

write(30,*)' Prandtl Number ',pr

write(30,*)' Hydro Relax. Time ',tauf

write(30,*)' Termo Relax. Time ',taug

write(30,*)' Solution Converge at ',nstep

write(30,*)' Delta t ',delt

write(30,*)' Mesh Size ', lx, ly

do j = 1,ly

write(30,*) u((lx+1)/2,j)*(ly-1)/chi

end do

close(30)

open(unit=31,file='vvel1.dat',status='replace',action='write',iosta

t=ierror)

do i = 1,lx

write(31,*) v(i,(ly+1)/2)*(ly-1)/chi

end do

close(31)

open(unit=32,file='variables.dat',status='replace',action='write',i

ostat=ierror)

write(32,*)'x-vel, y-vel, temp'

do j = 1,ly

do i = 1,lx

write(32,*) u(i,j)*(ly-1)/chi,v(i,j)*(ly-1)/chi,temp(i,j)

end do

end do

close(32)

stop

end

==================================================================

subroutine initial

49

===================================================================














common /con2/ lx,ly



g,rtaug




real w(0:kkk)

delx = 1.0

dely = 1.0

dely2 = dely*dely

delx2 = delx*delx

!setup physical data!

cx(0) = 0.0d0

cy(0) = 0.0d0

do k = 1,8

w(k) = sqrt (2.0d0)

if(mod(k,2) .eq. 1) w(k) = 1.0d0

cx(k) = w(k)*cos((k-1)*pi/4.0d0)

cy(k) = w(k)*sin((k-1)*pi/4.0d0)

end do

dx(1) = 1.0d0

dy(1) = 1.0d0

dx(2) = -1.0d0

dy(2) = 1.0d0

dx(3) = -1.0d0

dy(3) = -1.0d0

dx(4) = 1.0d0

dy(4) = -1.0d0

do i = 1,lx

do j = 1,ly

rho(i,j) = rhoint

if (i.eq.1) then

u(i,j) = 0.0

v(i,j) = 0.0

50

temp(i,j) = th

else

u(i,j) = 0.0

v(i,j) = 0.0

temp(i,j) = tc

end if

end do

end do

call equilibrium

do i = 0,lx+1

do j = 0,ly+1

do k = 0,8

f(i,j,k) = feq(i,j,k)

fx(i,j,k) =0.0

fy(i,j,k) =0.0

end do

end do

end do

do i = 0,lx+1

do j = 0,ly+1

do k = 1,4

g(i,j,k) = geq(i,j,k)

gx(i,j,k) =0.0

gy(i,j,k) =0.0

end do

end do

end do

return

end

===================================================================

subroutine equilibrium

===================================================================














common /con2/ lx,ly


common/con4/

pi,tauf,rhoint,delx,dely,delx2,dely2,rtauf,xnyu,chi,taug,rtaug

51




real u2(0:ij,0:ij)

do i = 1,lx

do j = 1,ly

u2(i,j) = u(i,j)**2 + v(i,j)**2

feq(i,j,0) = rho(i,j)*(1.0 - 3.0/2.0*u2(i,j))*4.0/9.0

do l = 1,4

k = l*2 ; dir = cx(k)*u(i,j) + cy(k)*v(i,j)

feq(i,j,k) = rho(i,j)*(1. + 3.*dir + 9./2.*dir**2 -

3./2.*u2(i,j))/36.

k = l*2 - 1; dir = cx(k)*u(i,j) + cy(k)*v(i,j)

feq(i,j,k) = rho(i,j)*(1. + 3.*dir + 9./2.*dir**2 -

3./2.*u2(i,j))/9.

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 1,4

tmpg = dx(k)*u(i,j) + dy(k)*v(i,j)

geq(i,j,k) = rho(i,j)*temp(i,j)*(1 + tmpg )/4

end do

end do

end do

return

end

==================================================================

subroutine fin

==================================================================



common/var1/

f(0:ij,0:ij,0:kkk),feq(0:ij,0:ij,0:kkk),g(0:ij,0:ij,0:kkk),geq(0:ij

,0:ij,0:kkk)










common /con2/ lx,ly


common/con4/

pi,tauf,rhoint,delx,dely,delx2,dely2,rtauf,xnyu,chi,taug,rtaug




52

totalf = 0.0

if (nstep .ne. 1) then

do j = 1,ly

do k = 0,8

f(0,j,k) = 2.0*f(1,j,k) - f(2,j,k)

f(lx+1,j,k) = 2.0*f(lx,j,k) - f(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 0,8

f(i,0,k) = 2.0*f(i,1,k) - f(i,2,k)

f(i,ly+1,k) = 2.0*f(i,ly,k) - f(i,ly-1,k)

end do

end do

do k = 0,8

f(0,0,k) = 2.0*f(1,1,k) - f(2,2,k)

f(0,ly+1,k) = 2.0*f(1,ly,k) - f(2,ly-1,k)

f(lx+1,0,k) = 2.0*f(lx,1,k) - f(lx-1,2,k)

f(lx+1,ly+1,k) = 2.0*f(lx,ly,k) - f(lx-1,ly-1,k)

end do

do j = 1,ly

do k = 0,8

fx(0,j,k) = 2.0*fx(1,j,k) - fx(2,j,k)

fx(lx+1,j,k) = 2.0*fx(lx,j,k) - fx(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 0,8

fx(i,0,k) = 2.0*fx(i,1,k) - fx(i,2,k)

fx(i,ly+1,k) = 2.0*fx(i,ly,k) - fx(i,ly-1,k)

end do

end do

do k = 0,8

fx(0,0,k) = 2.0*fx(1,1,k) - fx(2,2,k)

fx(0,ly+1,k) = 2.0*fx(1,ly,k) - fx(2,ly-1,k)

fx(lx+1,0,k) = 2.0*fx(lx,1,k) - fx(lx-1,2,k)

fx(lx+1,ly+1,k = 2.0*fx(lx,ly,k) - fx(lx-1,ly-1,k)

end do

do j = 1,ly

do k = 0,8

fy(0,j,k) = 2.0*fy(1,j,k) - fy(2,j,k)

fy(lx+1,j,k) = 2.0*fy(lx,j,k) - fy(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 0,8

53

fy(i,0,k) = 2.0*fy(i,1,k) - fy(i,2,k)

fy(i,ly+1,k) = 2.0*fy(i,ly,k) - fy(i,ly-1,k)

end do

end do

do k = 0,8

fy(0,0,k) = 2.0*fy(1,1,k) - fy(2,2,k)

fy(0,ly+1,k) = 2.0*fy(1,ly,k) - fy(2,ly-1,k)

fy(lx+1,0,k) = 2.0*fy(lx,1,k) - fy(lx-1,2,k)

fy(lx+1,ly+1,k) =2.0*fy(lx,ly,k) - fy(lx-1,ly-1,k)

end do

do j = 1,ly

do k = 0,8

feq(0,j,k) = 2.0*feq(1,j,k) - feq(2,j,k)

feq(lx+1,j,k)=2.0*feq(lx,j,k)- feq(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 0,8

feq(i,0,k) = 2.0*feq(i,1,k) - feq(i,2,k)

feq(i,ly+1,k)=2.0*feq(i,ly,k) -feq(i,ly-1,k)

end do

end do

do k = 0,8

feq(0,0,k) = 2.0*feq(1,1,k) - feq(2,2,k)

feq(0,ly+1,k) = 2.0*feq(1,ly,k) - feq(2,ly-1,k)

feq(lx+1,0,k) = 2.0*feq(lx,1,k) - feq(lx-1,2,k)

feq(lx+1,ly+1,k)=2.0*feq(lx,ly,k)-feq(lx-1,ly-1,k)

end do

end if

==================================================================

if (nstep .ne. 1) then

do j = 1,ly

do k = 1,4

g(0,j,k) = 2.0*g(1,j,k) - g(2,j,k)

g(lx+1,j,k) = 2.0*g(lx,j,k) - g(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 1,4

g(i,0,k) = 2.0*g(i,1,k) - g(i,2,k)

g(i,ly+1,k) = 2.0*g(i,ly,k) - g(i,ly-1,k)

end do

end do

do k = 1,4

g(0,0,k) = 2.0*g(1,1,k) - g(2,2,k)

g(0,ly+1,k) = 2.0*g(1,ly,k) - g(2,ly-1,k)

g(lx+1,0,k) = 2.0*g(lx,1,k) - g(lx-1,2,k)

g(lx+1,ly+1,k) = 2.0*g(lx,ly,k) - g(lx-1,ly-1,k)

54

end do

do j = 1,ly

do k = 1,4

gx(0,j,k) = 2.0*gx(1,j,k) - gx(2,j,k)

gx(lx+1,j,k) = 2.0*gx(lx,j,k) - gx(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 1,4

gx(i,0,k) = 2.0*gx(i,1,k) - gx(i,2,k)

gx(i,ly+1,k) = 2.0*gx(i,ly,k) - gx(i,ly-1,k)

end do

end do

do k = 1,4

gx(0,0,k) = 2.0*gx(1,1,k) - gx(2,2,k)

gx(0,ly+1,k) = 2.0*gx(1,ly,k) - gx(2,ly-1,k)

gx(lx+1,0,k) = 2.0*gx(lx,1,k) - gx(lx-1,2,k)

gx(lx+1,ly+1,k)=2.0*gx(lx,ly,k) - gx(lx-1,ly-1,k)

end do

do j = 1,ly

do k = 1,4

gy(0,j,k) = 2.0*gy(1,j,k) - gy(2,j,k)

gy(lx+1,j,k) = 2.0*gy(lx,j,k) - gy(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 1,4

gy(i,0,k) = 2.0*gy(i,1,k) - gy(i,2,k)

gy(i,ly+1,k) = 2.0*gy(i,ly,k) - gy(i,ly-1,k)

end do

end do

do k = 1,4

gy(0,0,k) = 2.0*gy(1,1,k) - gy(2,2,k)

gy(0,ly+1,k) = 2.0*gy(1,ly,k) - gy(2,ly-1,k)

gy(lx+1,0,k) = 2.0*gy(lx,1,k) - gy(lx-1,2,k)

gy(lx+1,ly+1,k) =2.0*gy(lx,ly,k) - gy(lx-1,ly-1,k)

end do

do j = 1,ly

do k = 1,4

geq(0,j,k) = 2.0*geq(1,j,k) - geq(2,j,k)

geq(lx+1,j,k =2.0*geq(lx,j,k) -geq(lx-1,j,k)

end do

end do

do i = 1,lx

do k = 1,4

geq(i,0,k) = 2.0*geq(i,1,k) - geq(i,2,k)

55

geq(i,ly+1,k) =2.0*geq(i,ly,k)-geq(i,ly-1,k)

end do

end do

do k = 1,4

geq(0,0,k) = 2.0*geq(1,1,k) - geq(2,2,k)

geq(0,ly+1,k) = 2.0*geq(1,ly,k) - geq(2,ly-1,k)

geq(lx+1,0,k) = 2.0*geq(lx,1,k) - geq(lx-1,2,k)

geq(lx+1,ly+1,k =2.0*geq(lx,ly,k)-geq(lx-1,ly-1,k)

end do

end if

do i = 1,lx

do j = 1,ly

do k = 0,8

fn(i,j,k)=f(i,j,k)-delt*rtauf*(f(i,j,k)-feq(i,j,k))

fxn(i,j,k)=fx(i,j,k)-

delt*rtauf*(fx(i,j,k)0.5*(feq(i+1,j,k)-feq(i-1,j,k)))

fyn(i,j,k)=fy(i,j,k)-delt*rtauf*(fy(i,j,k)-

0.5*(feq(i,j+1,k)-feq(i,j-1,k)))

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 0,8

f(i,j,k) = fn(i,j,k)

fx(i,j,k) = fxn(i,j,k)

fy(i,j,k) = fyn(i,j,k)

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 1,4

gn(i,j,k)=g(i,j,k)

delt*rtaug*(g(i,j,k)-geq(i,j,k))

gxn(i,j,k)=gx(i,j,k)-

delt*rtaug*(gx(i,j,k)- 0.5*(geq(i+1,j,k)-geq(i-1,j,k)))

gyn(i,j,k)=gy(i,j,k)-

delt*rtaug*(gy(i,j,k)- 0.5*(geq(i,j+1,k)-geq(i,j-1,k)))

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 1,4

g(i,j,k) = gn(i,j,k)

gx(i,j,k) = gxn(i,j,k)

gy(i,j,k) = gyn(i,j,k)

56

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 0,8

xx = -cx(k)*delt

yy = -cy(k)*delt

zx = sign(1.0,cx(k))

zy = sign(1.0,cy(k))

iup = i-int(zx)

jup = j-int(zy)

a1 = ((fx(iup,j,k) + fx(i,j,k))*delx*zx - 2.0*(f(i,j,k)

- f(iup,j,k)))/(delx**3*zx)

e1 = (3.0*(f(iup,j,k) - f(i,j,k)) + (fx(iup,j,k) +

2.*fx(i,j,k))*delx*zx)/(delx*delx)

b1 = ((fy(i,jup,k) + fy(i,j,k))*dely*zy - 2.0*(f(i,j,k)

- f(i,jup,k)))/(dely**3*zy)

f1 = (3.0*(f(i,jup,k) - f(i,j,k)) + (fy(i,jup,k) +

2.0*fy(i,j,k))*dely*zy)/dely**2

d1 = ( - (f(i,j,k) - f(i,jup,k) - f(iup,j,k) +

f(iup,jup,k)) - (fy(iup,j,k) -

fy(i,j,k))*dely*zy)/(delx*dely**2*zx)

c1 = ( - (f(i,j,k) - f(i,jup,k) - f(iup,j,k) +

f(iup,jup,k)) - (fx(i,jup,k) -

fx(i,j,k))*delx*zx)/(delx**2*dely*zy)

g1 = ( - (fy(iup,j,k) - fy(i,j,k)) +

c1*delx*delx)/(delx*zx)

fn(i,j,k) = ((a1*xx+c1*yy+e1)*xx + g1*yy + fx(i,j,k))*xx

+ ((b1*yy+d1*xx+f1)*yy + fy(i,j,k))*yy + f(i,j,k)

fxn(i,j,k) = (3.0*a1*xx + 2.0*(c1*yy+e1))*xx +

(d1*yy+g1)*yy+fx(i,j,k)

fyn(i,j,k) = (3.0*b1*yy + 2.0*(d1*xx+f1))*yy +

(c1*xx+g1)*xx+fy(i,j,k)

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 0,8

f(i,j,k) = fn(i,j,k)

fx(i,j,k) = fxn(i,j,k)

fy(i,j,k) = fyn(i,j,k)

if (f(i,j,k) <= 0 ) then

write (*,*) ' error'

end if

totalf = totalf + f(i,j,k)

end do

end do

end do

57

do i = 1,lx

do j = 1,ly

do k = 1,4

xx = -dx(k)*delt

yy = -dy(k)*delt

zx = sign(1.0,dx(k))

zy = sign(1.0,dy(k))

iup = i-int(zx)

jup = j-int(zy)

a1 = ((gx(iup,j,k) + gx(i,j,k))*delx*zx - 2.0*(g(i,j,k)

- g(iup,j,k)))/(delx**3*zx)

e1 = (3.0*(g(iup,j,k) - g(i,j,k)) + (gx(iup,j,k) +

2.*gx(i,j,k))*delx*zx)/(delx*delx)

b1 = ((gy(i,jup,k) + gy(i,j,k))*dely*zy - 2.0*(g(i,j,k)

- g(i,jup,k)))/(dely**3*zy)

f1 = (3.0*(g(i,jup,k) - g(i,j,k)) + (gy(i,jup,k) +

2.0*gy(i,j,k))*dely*zy)/dely**2

d1 = ( - (g(i,j,k) - g(i,jup,k) - g(iup,j,k) +

g(iup,jup,k)) - (gy(iup,j,k) -

gy(i,j,k))*dely*zy)/(delx*dely**2*zx)

c1 = ( - (g(i,j,k) - g(i,jup,k) - g(iup,j,k) +

g(iup,jup,k)) - (gx(i,jup,k) -

gx(i,j,k))*delx*zx)/(delx**2*dely*zy)

g1 = (- (gy(iup,j,k) - gy(i,j,k)) +

c1*delx*delx)/(delx*zx)

gn(i,j,k) = ((a1*xx+c1*yy+e1)*xx + g1*yy + gx(i,j,k))*xx

+ ((b1*yy+d1*xx+f1)*yy + gy(i,j,k))*yy + g(i,j,k)

gxn(i,j,k) = (3.0*a1*xx + 2.0*(c1*yy+e1))*xx +

(d1*yy+g1)*yy+gx(i,j,k)

gyn(i,j,k) = (3.0*b1*yy + 2.0*(d1*xx+f1))*yy +

(c1*xx+g1)*xx+gy(i,j,k)

end do

end do

end do

do i = 1,lx

do j = 1,ly

do k = 1,4

g(i,j,k) = gn(i,j,k)

gx(i,j,k) = gxn(i,j,k)

gy(i,j,k) = gyn(i,j,k)

if (g(i,j,k) <= -0.1 ) then

write (*,*) ' error',i,j,k,g(i,j,k)

end if

end do

end do

end do

if ( mod(nstep,ntin) == 0) then

58

write (*,60) totalf

60 format ('totalf = ',F8.2,//)

write (*,70) totalg

70 format ('totalg = ',F8.2,//)

end if

tempor = 0.0

do i = 1,lx

do j = 1,ly

tempor = tempor + temp(i,j)

tempave = tempor/(lx*ly)

end do

end do

do i = 1,lx

do j = 1,ly

do k = 0,8

f(i,j,k) = f(i,j,k)+3*delt*gra*(cy(k)-

v(i,j))*feq(i,j,k)*(temp(i,j)-tempave)

end do

end do

end do

return

end

==================================================================

subroutine output

==================================================================














common /con2/ lx,ly



g,rtaug




do i = 1,lx

do j = 1,ly

rho(i,j) = 0.0

end do

end do

do i = 2,lx-1

59

do j = 2,ly-1

do k = 0,8

rho(i,j) = rho(i,j) + f(i,j,k)

end do

end do

end do

do i = 2,lx-1

rho(i,1) = rho(i,2)

rho(i,ly) = rho(i,ly-1)

end do

do j = 2,ly-1

rho(1,j) = rho(2,j)

rho(lx,j) = rho(lx-1,j)

end do

rho(1,1) = rho(2,2)

rho(lx,1) = rho(lx-1,2)

rho(lx,ly) = rho(lx-1,ly-1)

rho(1,ly) = rho(2,ly-1)

do i= 2,lx-1

do j = 2,ly-1

u(i,j) = 0.d0

v(i,j) = 0.d0

temp(i,j) = 0.d0

end do

end do

do i = 2, lx-1

do j = 2, ly-1

do k = 0,8

u(i,j)=u(i,j)+ f(i,j,k)*cx(k)/rho(i,j)

v(i,j)=v(i,j)+ f(i,j,k)*cy(k)/rho(i,j)

end do

do k = 1,4

temp(i,j)=temp(i,j)+ g(i,j,k)/rho(i,j)

end do

end do

end do

do j = 1, ly

temp(1,j) = th

temp(lx,j) = tc

end do

do i = 1, lx

temp(i,ly) = temp(i,ly-1)

temp(i,1) = temp(i,2)

end do

return

end

60

APPENDIX C

MESH FOR 2D NATURAL CONVECTION

PARAMETER (NNN=230000, chi = 0.104519187198994)

parameter (xd = 51, yd = 51)

IMPLICIT REAL*8(A-H,O-Z)

integer node,ok1,ok2

DIMENSION U(NNN),V(NNN),temp(nnn)

DIMENSION Uu(101,101),tempp(101,101),vv(101,101)

open(unit=10,file='nd_number.dat',status='replace',action='write',i

ostat=ierror)

node = 0

do j = 1,yd

do i = 1,xd

node = node + 1

write(10,*) i,j,node

end do

end do

close(10)

open(unit=11,file='variables.dat',status='old',action='read',iostat

=ierror)

read(11,*)

do n = 1,node

read(11,*)u(n),v(n),temp(n)

end do

close(11)

open(unit=11,file='variables.dat',status='old',action='read',iostat

=ierror)

read(11,*)

do j = 1,yd

do i = 1,xd

read(11,*)uu(i,j),vv(i,j),tempp(i,j)

end do

end do

close(11)

do j = 1,yd

do i = 1,xd

uu(i,j) =uu(i,j)*chi/(yd-1)

end do

end do

umax = abs(u(1))

do i = 2, node

ux = abs(u(i))

if(ux .gt. umax) then

umax = ux

write(*,*)'node= ',i,'umax = ',umax

end if

end do

write(*,*)'next for v?'

read (*,*) ok1

vmax = abs(v(1))

do i = 2, node

uy = abs(v(i))

if(uy .gt. vmax) then

vmax = uy

61

write(*,*)'node= ',i,'vmax = ',vmax

end if

end do

! calculate nusselt number

th = 1.0d0

tc = 0.0d0

bnu = 0.0

do i = 1, xd

do j = 1,yd

if (i .eq. 1) then

bnu=bnu+(((xd-1)/(chi*(th-tc)*(xd)*(yd)))*((uu(i,j)*tempp(i,j))-

(chi*(tempp(i+1,j)-tempp(i,j)))))

else if (i .eq. xd) then

bnu=bnu+(((xd-1)/(chi*(thtc)*(xd)*(yd)))*((uu(i,j)*tempp(i,j))-

(chi*(tempp(i,j)-tempp(i-1,j)))))

else

bnu=bnu+(((xd-1)/(chi*(thtc)*(xd)*(yd)))*((uu(i,j)*tempp(i,j))-

(chi*(0.5*(tempp(i+1,j)-tempp(i-1,j))))))

end if

end do

end do

write(*,*) 'nusselt number = ',bnu

STOP

END

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